WEBVTT
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OK, this is the lecture on
positive definite matrices.
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I made a start on those
briefly in a previous lecture.
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One point I wanted to make was
the way that this topic brings
00:00:26.180 --> 00:00:30.300
the whole course together,
pivots, determinants,
00:00:30.300 --> 00:00:36.190
eigenvalues, and something
new- four plot instability
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and then something new in this
expression, x transpose Ax,
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actually that's the guy
to watch in this lecture.
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So, so the topic is
positive definite matrix,
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and what's my goal?
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First, first goal is, how
can I tell if a matrix is
00:00:58.440 --> 00:00:59.970
positive definite?
00:00:59.970 --> 00:01:02.490
So I would like to
have tests to see
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if you give me a, a
five by five matrix,
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how do I tell if it's
positive definite?
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More important is,
what does it mean?
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Why are we so interested
in this property
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of positive definiteness?
00:01:16.440 --> 00:01:21.410
And then, at the end
comes some geometry.
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Ellipses are connected with
positive definite things.
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Hyperbolas are not connected
with positive definite things,
00:01:28.520 --> 00:01:33.350
so we- it's this, we,
there's a geometry too,
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but mostly it's
linear algebra and --
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this application of how do you
recognize 'em when you have
00:01:43.020 --> 00:01:45.181
a minim is pretty neat.
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OK.
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I'm gonna begin with two by two.
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All matrices are
symmetric, right?
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That's understood; the
matrix is symmetric,
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now my question is, is
it positive definite?
00:02:00.650 --> 00:02:04.030
Now, here are some --
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each one of these is a complete
test for positive definiteness.
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If I know the eigenvalues,
my test is are they positive?
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Are they all positive?
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If I know these --
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so, A is really --
00:02:24.170 --> 00:02:27.570
I look at that number A,
here, as the, as the one
00:02:27.570 --> 00:02:33.420
by one determinant, and here's
the two by two determinant.
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So this is the determinant test.
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This is the eigenvalue test,
this is the determinant test.
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Are the determinants growing in
s- of all, of all end, sort of,
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can I call them
leading submatrices,
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they're the first
ones the northwest,
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Seattle submatrices coming
down from from there,
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they all, all those determinants
have to be positive,
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and then another
test is the pivots.
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The pivots of a
two by two matrix
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are the number A for sure,
and, since the product
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is the determinant,
the second pivot
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must be the determinant
divided by A.
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And then in here is gonna come
my favorite and my new idea,
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the, the, the the one to
catch, about x transpose Ax
00:03:30.450 --> 00:03:32.640
being positive.
00:03:32.640 --> 00:03:35.280
But we'll have to
look at this guy.
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This gets, like a star, because
for most, presentations,
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the definition of
positive definiteness
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would be this number four and
these numbers one two three
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would be test four.
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OK.
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Maybe I'll tuck this,
where, you know,
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OK.
00:03:56.660 --> 00:04:00.000
So I'll have to look
at this x transpose Ax.
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Can you, can we
just be sure, how
00:04:07.760 --> 00:04:13.130
do we know that the eigenvalue
test and the determinant test,
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pick out the same
matrices, and let me,
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let's just do a few examples.
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Some examples.
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Let me pick the matrix
two, six, six, tell me,
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what number do I have to
put there for the matrix
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to be positive definite?
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Tell me a sufficiently
large number
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that would make it
positive definite?
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Let's just practice with
these conditions in the two
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by two case.
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Now, when I ask
you that, you don't
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wanna find the eigenvalues, you
would use the determinant test
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for that, so, the first
or the pivot test,
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that, that guy is certainly
positive, that had to happen,
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and it's OK.
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How large a number here -- the
number had better be more than
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what?
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More than eighteen, right,
because if it's eight --
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no.
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More than what?
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Nineteen, is it?
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If I have a nineteen here,
is that positive definite?
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I get thirty eight minus
thirty six, that's OK.
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If I had an eighteen, let
me play it really close.
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If I have an eighteen there,
then I positive definite?
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Not quite.
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I would call this
guy positive, so it's
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useful just to see that
this the borderline.
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That matrix is on
the borderline,
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I would call that matrix
positive semi-definite.
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And what are the
eigenvalues of that matrix,
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just since we're given
eigenvalues of two by twos,
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when it's semi-definite, but
not definite, then the --
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I'm squeezing this
eigenvalue test down, --
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what's the eigenvalue that
I know this matrix has?
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What kind of a matrix
have I got here?
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It's a singular matrix, one
of its eigenvalues is zero.
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That has an eigenvalue zero,
and the other eigenvalue is --
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from the trace, twenty.
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OK.
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So that, that matrix has
eigenvalues greater than
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or equal to zero, and it's that
"equal to" that brought this
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word "semi-definite" in.
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And, the what are the
pivots of that matrix?
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So the pivots, so
the eigenvalues
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are zero and twenty, the pivots
are, well, the pivot is two,
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and what's the next pivot?
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There isn't one.
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We got a singular matrix here,
it'll only have one pivot.
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You see that that's a rank
one matrix, two six is a --
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six eighteen is a
multiple of two six,
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the matrix is singular
it only has one pivot,
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so the pivot test doesn't quite
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The -- let me do
the x transpose Ax.
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pass.
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Now this is --
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the novelty now.
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OK.
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What I looking at when I
look at this new combination,
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x transpose Ax.
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x is any vector now, so
lemme compute, so any vector,
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lemme call its components
x1 and x2, so that's x.
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And I put in here A.
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Let's, let's use this example
two six, six eighteen,
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and here's x
transposed, so x1 x2.
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We're back to real matrices,
after that last lecture
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that- that said what to do
in the complex case, let's
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come back to real matrices.
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So here's x transpose Ax,
and what I'm interested
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in is, what do I get if I
multiply those together?
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I get some function of x1
and x2, and what is it?
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Let's see, if I do this
multiplication, so I do it,
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lemme, just, I'll just
do it slowly, x1, x2,
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if I multiply that matrix,
this is 2x1, and 6x2s,
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and the next row
is 6x1s and 18x2s,
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and now I do this final
step and what do I have?
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I've got 2x1 squareds,
got 2X1 squareds
00:09:16.810 --> 00:09:23.080
is coming from this two, I've
got this one gives me eighteen,
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well, shall I do the
ones in the middle?
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How many x1 x2s do I have?
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Let's see, x1 times that
6x2 would be six of 'em,
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and then x2 times this one
will be six more, I get twelve.
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So, in here is going, this
is the number a, this is
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the number 2b, and in here is --
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x2 times eighteen x2 will be
eighteen x2 squareds and this
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is the number c.
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So it's ax1 -- it's
like ax squared.
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2bxy and cy squared.
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For my first point that I
wanted to make by just doing out
00:10:11.280 --> 00:10:15.720
a multiplication is, that is as
soon as you give me the matrix,
00:10:15.720 --> 00:10:19.240
as soon as you give me
the matrix, I can --
00:10:19.240 --> 00:10:22.860
those are the numbers
that appear in --
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I'll call this
thing a quadratic,
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you see, it's not
linear anymore.
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Ax is linear, but now I've
got an x transpose coming
00:10:32.360 --> 00:10:35.680
in, I'm up to degree
two, up to degree two,
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maybe quadratic is the --
00:10:38.880 --> 00:10:39.480
use the word.
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A quadratic form.
00:10:41.980 --> 00:10:46.610
It's purely degree two,
there's no linear part,
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there's no constant
part, there certainly
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no cubes or fourth powers,
it's all second degree.
00:10:53.540 --> 00:10:55.780
And my question is --
00:10:55.780 --> 00:11:00.120
is that quantity
positive or not?
00:11:00.120 --> 00:11:08.250
That's -- for every x1 and x2,
that is my new definition --
00:11:08.250 --> 00:11:11.920
that's my definition of a
positive definite matrix.
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If this quantity is positive,
if, if, if, it's positive
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for all x's and y's, all
x1 x2s, then I call them --
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then that's the matrix
is positive definite.
00:11:29.170 --> 00:11:34.340
Now, is this guy
passing our test?
00:11:34.340 --> 00:11:38.120
Well we have, we anticipated
the answer here by,
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by asking first about
eigenvalues and pivots,
00:11:42.420 --> 00:11:44.820
and what happened?
00:11:44.820 --> 00:11:46.350
It failed.
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It barely failed.
00:11:49.490 --> 00:11:53.670
If I had made this
eighteen down to a seven,
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it would've totally failed.
00:11:58.120 --> 00:12:01.540
I do that with the eraser, and
then I'll put back eighteen,
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because, seven is such a
total disaster, but if --
00:12:07.170 --> 00:12:10.240
I'll keep seven for a second.
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Is that thing in any
way positive definite?
00:12:15.170 --> 00:12:18.480
No, absolutely not.
00:12:18.480 --> 00:12:21.990
I don't know its eigenvalues,
but I know for sure one of them
00:12:21.990 --> 00:12:24.230
is negative.
00:12:24.230 --> 00:12:28.940
Its pivots are two and then
the next pivot would be
00:12:28.940 --> 00:12:32.250
the determinant over two, and
the determinant is -- what,
00:12:32.250 --> 00:12:35.090
what's the determinant
of this thing?
00:12:35.090 --> 00:12:37.620
Fourteen minus
thirty six, I've got
00:12:37.620 --> 00:12:39.820
a determinant minus twenty two.
00:12:39.820 --> 00:12:43.330
The next pivot will
be -- the pivots now,
00:12:43.330 --> 00:12:48.480
of this thing are two and
minus eleven or something.
00:12:48.480 --> 00:12:51.320
Their product being minus
twenty two the determinant.
00:12:51.320 --> 00:12:53.620
This thing is not
positive definite.
00:12:53.620 --> 00:12:57.200
What would be -- let me look
at the x transpose Ax for this
00:12:57.200 --> 00:12:57.700
guy.
00:12:57.700 --> 00:13:00.460
What's -- if I do out
this multiplication,
00:13:00.460 --> 00:13:04.560
this eighteen is temporarily
changing to a seven.
00:13:04.560 --> 00:13:08.130
This eighteen is temporarily
changing to a seven,
00:13:08.130 --> 00:13:16.610
and I know that there's
some numbers x1 and x2
00:13:16.610 --> 00:13:24.880
for which that thing, that
function, is negative.
00:13:24.880 --> 00:13:28.640
And I'm desperately trying
to think what they are.
00:13:28.640 --> 00:13:30.040
Maybe you can see.
00:13:30.040 --> 00:13:33.940
Can you tell me a
value of x1 and x2
00:13:33.940 --> 00:13:36.675
that makes this
quantity negative?
00:13:40.400 --> 00:13:43.180
Oh, maybe one and minus one?
00:13:43.180 --> 00:13:49.270
Yes, that's -- in this case,
that, will work, right,
00:13:49.270 --> 00:13:54.150
if I took x1 to be one,
and x2 to be minus one,
00:13:54.150 --> 00:13:57.170
then I always get something
positive, the two,
00:13:57.170 --> 00:14:01.367
and the seven minus one squared,
but this would be minus twelve
00:14:01.367 --> 00:14:02.950
and the whole thing
would be negative;
00:14:02.950 --> 00:14:07.620
I would have two minus twelve
plus seven, a negative.
00:14:07.620 --> 00:14:11.550
If I drew the graph, can I
get the little picture in
00:14:11.550 --> 00:14:12.050
here?
00:14:12.050 --> 00:14:16.430
If I draw the graph
of this thing?
00:14:16.430 --> 00:14:22.730
So, graphs, of the
function f(x,y), or f(x),
00:14:22.730 --> 00:14:29.620
so I say here f(x,y) equal
this -- x transpose Ax, this,
00:14:29.620 --> 00:14:32.040
this this ax squared,
2bxy, and cy squared.
00:14:32.040 --> 00:14:46.600
And, let's take the
example, with these numbers.
00:14:49.210 --> 00:14:54.650
OK, so here's the x axis,
here's the y axis, and here's --
00:14:54.650 --> 00:14:56.380
up is the function.
00:14:56.380 --> 00:14:59.390
z, if you like, or f.
00:14:59.390 --> 00:15:03.940
I apologize, and let me,
just once in my life here,
00:15:03.940 --> 00:15:08.780
put an arrow over these,
these, axes so you see them.
00:15:08.780 --> 00:15:13.300
That's the vector and I just,
see, instead of x1 and x2,
00:15:13.300 --> 00:15:16.220
I made them x- the
components x and y.
00:15:16.220 --> 00:15:16.930
OK.
00:15:16.930 --> 00:15:23.660
So, so, what's a graph of 2x
squared, twelve xy, and seven y
00:15:23.660 --> 00:15:25.710
squared?
00:15:25.710 --> 00:15:27.660
I'd like to see --
00:15:27.660 --> 00:15:31.260
I not the greatest
artist, but let's --
00:15:31.260 --> 00:15:38.400
tell me something about
this graph of this function.
00:15:42.560 --> 00:15:44.740
Whoa, tell me one point
that it goes through.
00:15:47.970 --> 00:15:49.470
The origin.
00:15:49.470 --> 00:15:50.990
Right?
00:15:50.990 --> 00:15:56.450
Even this artist can get this
thing to go through the origin,
00:15:56.450 --> 00:16:00.421
when these are zero, I,
I certainly get zero.
00:16:00.421 --> 00:16:00.920
OK.
00:16:00.920 --> 00:16:02.540
Some more points.
00:16:02.540 --> 00:16:06.570
If x is one and y is zero,
then I'm going upwards,
00:16:06.570 --> 00:16:09.060
so I'm going up this
way, and I'm, I'm
00:16:09.060 --> 00:16:12.180
going up, like, two x
squared in that direction.
00:16:12.180 --> 00:16:14.766
So -- that's meant
to be a parabola.
00:16:18.200 --> 00:16:23.150
And, suppose x stays
zero and y increases.
00:16:23.150 --> 00:16:26.760
Well, y could be positive or
negative; it's seven y squared.
00:16:26.760 --> 00:16:30.530
Is this function going upward?
00:16:30.530 --> 00:16:35.640
In the x direction it's going
upward, and in the y direction
00:16:35.640 --> 00:16:39.900
it's going upwards,
and if x equals y
00:16:39.900 --> 00:16:42.140
then the forty-five degree
direction is certainly
00:16:42.140 --> 00:16:44.810
going upwards; because
then we'd have what,
00:16:44.810 --> 00:16:49.600
about, everything would
be positive, but what?
00:16:49.600 --> 00:16:54.810
This function -- what's
the graph of this function?
00:16:54.810 --> 00:16:55.310
Look like?
00:16:55.310 --> 00:17:00.450
Tell me the word that describes
the graph of this function.
00:17:00.450 --> 00:17:02.980
This is the non-positive
definite here,
00:17:02.980 --> 00:17:07.660
everybody's with me
here, for some reason
00:17:07.660 --> 00:17:10.740
got started in a negative
direction, your case that
00:17:10.740 --> 00:17:12.750
isn't positive definite.
00:17:12.750 --> 00:17:17.190
And what's the graph look like
that goes up, but does it --
00:17:17.190 --> 00:17:22.710
do we have a minimum here, does
it go from, from the origin?
00:17:22.710 --> 00:17:24.500
Completely?
00:17:24.500 --> 00:17:28.710
No, because we just checked
that this thing failed.
00:17:28.710 --> 00:17:33.490
It failed along the direction
when x was minus y --
00:17:33.490 --> 00:17:37.080
we have a saddle point,
let me put myself, let me,
00:17:37.080 --> 00:17:39.350
to the least, tell you the word.
00:17:39.350 --> 00:17:46.470
This thing, goes up
in some directions,
00:17:46.470 --> 00:17:53.330
but down in other directions,
and if we actually knew what
00:17:53.330 --> 00:17:59.300
a saddle looked like or
thinks saddles do that --
00:17:59.300 --> 00:18:06.560
the way your legs go is,
like, down, up, the way, you,
00:18:06.560 --> 00:18:16.200
looking like, forward, and, the,
and drawing the thing is even
00:18:16.200 --> 00:18:17.330
worse than describing --
00:18:17.330 --> 00:18:20.760
I'm just going to say in
some directions we go up
00:18:20.760 --> 00:18:28.290
and in other directions,
there is, a saddle --
00:18:28.290 --> 00:18:30.330
Now I'm sorry I put
that on the front board,
00:18:30.330 --> 00:18:34.450
you have no way to cover
it, but it's a saddle.
00:18:34.450 --> 00:18:35.360
OK.
00:18:35.360 --> 00:18:38.950
And, and this is a
saddle point, it's
00:18:38.950 --> 00:18:43.900
the, it's the point that's at
the maximum in some directions
00:18:43.900 --> 00:18:46.770
and at the minimum
in other directions.
00:18:46.770 --> 00:18:50.840
And actually, the perfect
directions to look
00:18:50.840 --> 00:18:53.670
are the eigenvector directions.
00:18:53.670 --> 00:18:55.260
We'll see that.
00:18:55.260 --> 00:19:03.730
So this is, not a
positive definite matrix.
00:19:03.730 --> 00:19:04.280
OK.
00:19:04.280 --> 00:19:08.190
Now I'm coming back
to this example,
00:19:08.190 --> 00:19:14.400
getting rid of this seven,
let's move it up to twenty.
00:19:14.400 --> 00:19:18.420
Let's, let's let's make the
thing really positive definite.
00:19:18.420 --> 00:19:19.470
OK.
00:19:19.470 --> 00:19:22.990
So this is, this
number's now twenty.
00:19:22.990 --> 00:19:24.360
c is now twenty.
00:19:24.360 --> 00:19:24.860
OK.
00:19:24.860 --> 00:19:30.800
Now that passes the test, which
I haven't proved, of course,
00:19:30.800 --> 00:19:34.960
it passes the test
for positive pivots.
00:19:34.960 --> 00:19:40.270
It passes the test for
positive eigenvalues.
00:19:40.270 --> 00:19:43.160
How can you tell that the
eigenvalues of that matrix
00:19:43.160 --> 00:19:45.850
are positive without
actually finding them?
00:19:45.850 --> 00:19:49.400
Of course, two by two I could
find them, but can you see --
00:19:49.400 --> 00:19:51.530
how do I know they're positive?
00:19:51.530 --> 00:19:53.270
I know that their product is --
00:19:53.270 --> 00:19:58.020
I know that lambda one times
lambda two is positive, why?
00:19:58.020 --> 00:20:01.750
Because that's the
determinant, right,
00:20:01.750 --> 00:20:06.000
lambda one times lambda two is
the determinant, which is forty
00:20:06.000 --> 00:20:07.730
minus thirty-six is four.
00:20:07.730 --> 00:20:11.120
So the determinant is four.
00:20:11.120 --> 00:20:16.160
And the trace, the sum down
the diagonal, is twenty-two.
00:20:16.160 --> 00:20:20.630
So, they multiply to give four.
00:20:20.630 --> 00:20:22.850
So that leaves the
possibility they're either
00:20:22.850 --> 00:20:25.730
both positive or both negative.
00:20:25.730 --> 00:20:29.979
But if they're both negative,
the trace couldn't be
00:20:29.979 --> 00:20:31.520
So they're both
positive. twenty-two.
00:20:31.520 --> 00:20:34.170
So both of the eigenvalues
that are positive,
00:20:34.170 --> 00:20:36.220
both of the pivots
are positive --
00:20:36.220 --> 00:20:39.820
the determinants are positive,
and I believe that this
00:20:39.820 --> 00:20:47.260
function is positive everywhere
except at zero, zero,
00:20:47.260 --> 00:20:48.240
of course.
00:20:48.240 --> 00:20:52.330
When I write down
this condition,
00:20:52.330 --> 00:20:54.670
So I believe that
x transposed, let
00:20:54.670 --> 00:21:00.280
me copy, x transpose Ax is
positive, except, of course,
00:21:00.280 --> 00:21:07.450
at the minimum point,
at zero, of course,
00:21:07.450 --> 00:21:08.830
I don't expect miracles.
00:21:11.760 --> 00:21:15.720
So what does its graph look
like, and how do I check,
00:21:15.720 --> 00:21:18.730
and how do I check that
this really is positive?
00:21:22.300 --> 00:21:24.960
So we take it's
graph for a minute.
00:21:24.960 --> 00:21:27.200
What would be the graph
of that function --
00:21:27.200 --> 00:21:29.400
it does not have a saddle point.
00:21:29.400 --> 00:21:31.280
Let me -- I'll raise
the board, here,
00:21:31.280 --> 00:21:33.320
and stay with this
example for a while.
00:21:36.280 --> 00:21:40.660
So I want to do the graph
of -- here's my function,
00:21:40.660 --> 00:21:46.940
two x squared, twelve xy-s, that
could be positive or negative,
00:21:46.940 --> 00:21:48.510
and twenty y squared.
00:21:48.510 --> 00:21:56.290
But my point is, so you're
seeing the underlying point is,
00:21:56.290 --> 00:21:59.910
that, the things are
positive definite
00:21:59.910 --> 00:22:05.730
when in some way, these,
these pure squares, squares
00:22:05.730 --> 00:22:10.180
we know to be positive, and
when those kind of overwhelm
00:22:10.180 --> 00:22:13.527
this guy, who could be
m- positive or negative,
00:22:13.527 --> 00:22:15.110
because some like
or have same or have
00:22:15.110 --> 00:22:19.760
same or different signs,
when these are big enough
00:22:19.760 --> 00:22:22.850
they overwhelm this guy and
make the total thing positive,
00:22:22.850 --> 00:22:25.750
and what would the
graph now look like?
00:22:25.750 --> 00:22:32.940
Let me draw the x - well, let
me draw the x direction, the y
00:22:32.940 --> 00:22:40.970
direction, and the origin,
at zero, zero, I'm there,
00:22:40.970 --> 00:22:45.650
where do I go as I move
away from the origin?
00:22:45.650 --> 00:22:50.780
Where do I go as I move
away from the origin?
00:22:50.780 --> 00:22:53.240
I'm sure that I go up.
00:22:53.240 --> 00:22:56.660
The origin, the
center point here,
00:22:56.660 --> 00:23:01.730
is a minim because this thing I
believe, and we better see why,
00:23:01.730 --> 00:23:07.620
it's, the graph is like a bowl,
the graph is like a bowl shape,
00:23:07.620 --> 00:23:09.805
it's -- here's the minimum.
00:23:15.660 --> 00:23:19.030
And because we've
got a pure quadratic,
00:23:19.030 --> 00:23:23.410
we know it sits at the origin,
we know it's tangent plane,
00:23:23.410 --> 00:23:30.970
the first derivatives are zero,
so, we know, first derivatives,
00:23:30.970 --> 00:23:37.410
First derivatives are
all zero, but that's
00:23:37.410 --> 00:23:38.560
not enough for a minimum.
00:23:38.560 --> 00:23:43.020
It's first derivatives
were zero here.
00:23:45.780 --> 00:23:51.200
So, the partial derivatives,
the first derivatives, are zero.
00:23:51.200 --> 00:23:56.720
Again, because first derivatives
are gonna have an x or an a y,
00:23:56.720 --> 00:23:59.730
or a y in them, they'll
be zero at the origin.
00:23:59.730 --> 00:24:03.340
It's the second derivatives
that control everything.
00:24:03.340 --> 00:24:07.810
It's the second derivatives
that this matrix is telling us,
00:24:07.810 --> 00:24:10.630
and somehow --
00:24:10.630 --> 00:24:11.780
here's my point.
00:24:11.780 --> 00:24:16.710
You remember in Calculus, how
did you decide on a minimum?
00:24:16.710 --> 00:24:20.010
First requirement was, that
the derivative had to be
00:24:20.010 --> 00:24:20.940
zero.
00:24:20.940 --> 00:24:25.890
But then you didn't know if
you had a minimum or a maximum.
00:24:25.890 --> 00:24:27.500
To know that you
had a minimum, you
00:24:27.500 --> 00:24:30.300
had to look at the
second derivative.
00:24:30.300 --> 00:24:33.400
The second derivative
had to be positive,
00:24:33.400 --> 00:24:36.900
the slope had to be
increasing as you
00:24:36.900 --> 00:24:40.010
went through the minimum point.
00:24:40.010 --> 00:24:43.470
The curvature had to
go upwards, and that's
00:24:43.470 --> 00:24:46.910
what we're doing now
in two dimensions,
00:24:46.910 --> 00:24:49.280
and in n dimensions.
00:24:49.280 --> 00:24:52.030
So we're doing what
we did in Calculus.
00:24:52.030 --> 00:24:55.150
Second derivative
positive, m- will now
00:24:55.150 --> 00:24:58.590
become that the matrix
of second derivatives
00:24:58.590 --> 00:25:00.810
is positive definite.
00:25:00.810 --> 00:25:02.650
Can I just --
00:25:02.650 --> 00:25:05.670
like a translation of --
00:25:05.670 --> 00:25:11.890
this is how minimum are coming
in, ithe beginning of Calculus
00:25:11.890 --> 00:25:14.640
--
00:25:14.640 --> 00:25:22.730
we had a minimum was associated
with second derivative,
00:25:22.730 --> 00:25:24.170
being positive.
00:25:24.170 --> 00:25:27.340
And first derivative
zero, of course.
00:25:27.340 --> 00:25:36.120
Derivative, first
derivative, but it
00:25:36.120 --> 00:25:39.870
was the second derivative
that told us we had a minimum.
00:25:39.870 --> 00:25:43.510
And now, in 18.06,
in linear algebra,
00:25:43.510 --> 00:25:47.260
we'll have a minim
for our function now,
00:25:47.260 --> 00:25:53.040
our function will have, for
your function be a function not
00:25:53.040 --> 00:26:00.290
of just x but several variables,
the way functions really
00:26:00.290 --> 00:26:03.370
are in real life,
the minimum will
00:26:03.370 --> 00:26:15.140
be when the matrix of second
derivatives, the matrix
00:26:15.140 --> 00:26:17.590
here was one by one, there was
just one second derivative,
00:26:17.590 --> 00:26:20.560
now we've got lots.
00:26:20.560 --> 00:26:25.335
Is positive definite.
00:26:29.070 --> 00:26:31.450
So positive for a
number translates
00:26:31.450 --> 00:26:34.450
into positive
definite for a matrix.
00:26:34.450 --> 00:26:38.860
And it this brings
everything you check pivots,
00:26:38.860 --> 00:26:41.680
you check determinants,
you check all your values,
00:26:41.680 --> 00:26:44.910
or you check this minimum stuff.
00:26:44.910 --> 00:26:45.410
OK.
00:26:45.410 --> 00:26:49.190
Let me come back to this graph.
00:26:49.190 --> 00:26:50.595
That graph goes upwards.
00:26:53.220 --> 00:26:54.740
And I'll have to see why.
00:26:54.740 --> 00:26:58.360
How do I know that this,
that this function is always
00:26:58.360 --> 00:26:59.980
positive?
00:26:59.980 --> 00:27:04.770
Can you look at that and tell
that it's always positive?
00:27:04.770 --> 00:27:08.710
Maybe two by two, you
could feel pretty sure,
00:27:08.710 --> 00:27:14.530
but what's the good way to
show that this thing is always
00:27:14.530 --> 00:27:20.630
If we can express it, as, in
terms of squares, positive?
00:27:20.630 --> 00:27:24.010
because that's what we
know for any x and y,
00:27:24.010 --> 00:27:26.760
whatever, if we're
squaring something
00:27:26.760 --> 00:27:29.470
we certainly are not negative.
00:27:29.470 --> 00:27:32.690
So I believe that this
expression, this function,
00:27:32.690 --> 00:27:37.190
could be written as
a sum of squares.
00:27:37.190 --> 00:27:41.170
Can you tell me --
00:27:41.170 --> 00:27:43.460
see, because all the
problems, the headaches
00:27:43.460 --> 00:27:47.230
are coming from this xy term.
00:27:47.230 --> 00:27:50.500
If we can get expressions
-- if we can get that inside
00:27:50.500 --> 00:27:53.660
a square, so actually, what
we're doing is something
00:27:53.660 --> 00:27:58.160
called, that you've seen
called completing the square.
00:27:58.160 --> 00:28:01.210
Let me start the square
and you complete it.
00:28:01.210 --> 00:28:05.820
OK, I think we
have two of x plus,
00:28:05.820 --> 00:28:09.400
now I don't remember how
many y-s we need, but you'll
00:28:09.400 --> 00:28:10.725
figure it out, squared.
00:28:13.750 --> 00:28:20.470
How many y-s should I
put in here, to make --
00:28:20.470 --> 00:28:23.950
what do I want to do, the two
x squared-s will be correct,
00:28:23.950 --> 00:28:25.770
right?
00:28:25.770 --> 00:28:28.990
What I want to do is put
in the right number of y-s
00:28:28.990 --> 00:28:33.270
to get the twelve xy correct.
00:28:33.270 --> 00:28:34.885
And what is that number of y-s?
00:28:37.430 --> 00:28:39.170
Let's see, I've got
two times, and so
00:28:39.170 --> 00:28:41.770
I really want six xy-s
to come out of here,
00:28:41.770 --> 00:28:44.650
I think maybe if
I put three there,
00:28:44.650 --> 00:28:48.590
does that look right to you?
00:28:48.590 --> 00:28:53.940
I have two- this is, we
can mentally, multiply out,
00:28:53.940 --> 00:28:56.420
that's X squared,
that's right, that's
00:28:56.420 --> 00:28:59.580
six X Y, times the
two gives from, right,
00:28:59.580 --> 00:29:02.660
and how many Y squareds
have I now got?
00:29:02.660 --> 00:29:05.950
How many Y squareds have
I now got from this term?
00:29:05.950 --> 00:29:07.750
Eighteen.
00:29:07.750 --> 00:29:12.000
Eighteen was the key
number, remember?
00:29:12.000 --> 00:29:16.630
Now if I want to make it
twenty, then I've got two left.
00:29:16.630 --> 00:29:18.100
Two y squared-s.
00:29:18.100 --> 00:29:25.660
That's completing the
square, and it's, now
00:29:25.660 --> 00:29:28.120
I can see that that
function is positive,
00:29:28.120 --> 00:29:30.090
because it's all squares.
00:29:33.340 --> 00:29:35.840
I've got two squares,
added together,
00:29:35.840 --> 00:29:38.110
I couldn't go negative.
00:29:38.110 --> 00:29:42.070
What if I went
back to that seven?
00:29:42.070 --> 00:29:45.080
If instead of twenty that
number was a seven, then
00:29:45.080 --> 00:29:47.610
what would happen?
00:29:47.610 --> 00:29:51.010
This would still be correct,
I'd still have this square,
00:29:51.010 --> 00:29:53.670
to get the two x squared
and the twelve xy,
00:29:53.670 --> 00:29:58.800
and I'd have eighteen y squared
and then what would I do here?
00:29:58.800 --> 00:30:03.760
I'd have to remove eleven
y squared-s, right,
00:30:03.760 --> 00:30:08.200
if I only had a seven
here, then instead of --
00:30:08.200 --> 00:30:13.040
when I had a twenty I had two
more to put in, when I had
00:30:13.040 --> 00:30:16.740
an eighteen, which
was the marginal case,
00:30:16.740 --> 00:30:19.780
I had no more to put in.
00:30:19.780 --> 00:30:23.370
When I had a seven, which
was the case below zero,
00:30:23.370 --> 00:30:29.870
the indefinite case,
I had minus eleven.
00:30:29.870 --> 00:30:36.360
Now, so, you can see now,
that this thing is a bowl.
00:30:36.360 --> 00:30:36.860
OK.
00:30:36.860 --> 00:30:42.200
It's going upwards, if I cut
it at a plane, z equal to one,
00:30:42.200 --> 00:30:47.830
say, I would get, I would
get a curve, what would
00:30:47.830 --> 00:30:50.280
be the equation for that curve?
00:30:50.280 --> 00:30:53.190
If I cut it at height
one, the equation
00:30:53.190 --> 00:30:56.980
would be this
thing equal to one.
00:30:56.980 --> 00:30:59.910
And that curve
would be an ellipse.
00:30:59.910 --> 00:31:02.560
So actually,
already, I've blocked
00:31:02.560 --> 00:31:09.380
into the lecture, the different
pieces that we're aiming for.
00:31:09.380 --> 00:31:12.620
We're aiming for the
tests, which this passed;
00:31:12.620 --> 00:31:17.360
we're aiming for the connection
to a minimum, which this --
00:31:17.360 --> 00:31:22.420
which we see in the graph,
and if we chop it up,
00:31:22.420 --> 00:31:25.180
if we set this
thing equal to one,
00:31:25.180 --> 00:31:27.560
if I set that thing
equal to one, that --
00:31:27.560 --> 00:31:30.490
what that gives me
is, the cross-section.
00:31:30.490 --> 00:31:37.750
It gives me this, this
curve, and its equation
00:31:37.750 --> 00:31:41.200
is this thing equals one,
and that's an ellipse.
00:31:41.200 --> 00:31:44.660
Whereas if I cut
through a saddle point,
00:31:44.660 --> 00:31:47.900
I get a hyperbola.
00:31:47.900 --> 00:31:53.820
But this minimum stuff is
really what I'm most interested
00:31:53.820 --> 00:31:54.610
OK.
00:31:54.610 --> 00:31:55.110
in.
00:31:55.110 --> 00:31:55.610
OK.
00:31:55.610 --> 00:32:00.300
By -- I just have to ask,
do you recognize, I mean,
00:32:00.300 --> 00:32:04.460
these numbers here, the
two that appeared outside,
00:32:04.460 --> 00:32:08.090
the three that appeared inside,
the two that appeared there --
00:32:08.090 --> 00:32:13.690
actually, those numbers
come from elimination.
00:32:13.690 --> 00:32:18.890
Completing the square
is our good old method
00:32:18.890 --> 00:32:24.910
of Gaussian elimination,
in expressed
00:32:24.910 --> 00:32:27.280
in terms of these squares.
00:32:27.280 --> 00:32:30.600
The -- let me show
you what I mean.
00:32:30.600 --> 00:32:34.050
I just think those
numbers are no accident,
00:32:34.050 --> 00:32:40.050
If I take my matrix two,
six, six, and twenty,
00:32:40.050 --> 00:32:45.050
and I do elimination,
then the pivot is two
00:32:45.050 --> 00:32:50.150
and I take three,
what's the multiplier?
00:32:50.150 --> 00:32:54.180
How much of row one do I
take away from row two?
00:32:54.180 --> 00:32:55.360
Three.
00:32:55.360 --> 00:32:58.700
So what you're seeing in
this, completing the square,
00:32:58.700 --> 00:33:05.180
is the pivots outside and
the multiplier inside.
00:33:05.180 --> 00:33:06.680
Just do that again?
00:33:06.680 --> 00:33:12.840
The pivot is two, three -- three
of those away from that gives
00:33:12.840 --> 00:33:18.100
me two, six, zero, and
what's the second pivot?
00:33:18.100 --> 00:33:21.550
Three of this away from this,
three sixes'll be eighteen,
00:33:21.550 --> 00:33:23.380
and the second
pivot will also be a
00:33:23.380 --> 00:33:24.040
two.
00:33:24.040 --> 00:33:32.490
So that's the U, this is
the A, and of course the L
00:33:32.490 --> 00:33:37.900
was one, zero, one, and
the multiplier was three.
00:33:40.540 --> 00:33:48.210
So, completing the
square is elimination.
00:33:48.210 --> 00:33:54.230
Why I happy to see, happy
to see that coming together?
00:33:54.230 --> 00:34:01.190
Because I know about
elimination for m by m matrices.
00:34:01.190 --> 00:34:07.740
I just started talking about
completing the square, here,
00:34:07.740 --> 00:34:10.320
for two by twos.
00:34:10.320 --> 00:34:12.989
But now I see what's going on.
00:34:12.989 --> 00:34:17.909
Completing the square really
amounts to splitting this thing
00:34:17.909 --> 00:34:21.610
into a sum of squares, so
what's the critical thing --
00:34:21.610 --> 00:34:25.380
I have a lot of squares,
and inside those squares
00:34:25.380 --> 00:34:28.020
are multipliers but
they're squares,
00:34:28.020 --> 00:34:31.790
and the question is, what's
outside these squares?
00:34:31.790 --> 00:34:35.210
When I complete the square,
what are the numbers that go
00:34:35.210 --> 00:34:36.179
outside?
00:34:36.179 --> 00:34:37.342
They're the pivots.
00:34:39.889 --> 00:34:45.510
They're the pivots, and that's
why positive pivots give me
00:34:45.510 --> 00:34:47.409
sum of squares.
00:34:47.409 --> 00:34:50.270
Positive pivots, those
pivots are the numbers
00:34:50.270 --> 00:34:53.600
that go outside the
squares, so positive pivots,
00:34:53.600 --> 00:34:57.850
sum of squares, everything
positive, graph goes up,
00:34:57.850 --> 00:35:03.050
a minimum at the origin,
it's all connected together;
00:35:03.050 --> 00:35:04.710
all connected together.
00:35:04.710 --> 00:35:08.110
And in the two by two case,
you can see those connections,
00:35:08.110 --> 00:35:16.100
but linear algebra now can go
up to three by three, m by m.
00:35:16.100 --> 00:35:17.950
Let's do that next.
00:35:17.950 --> 00:35:20.680
Can I just, before
I leave two by two,
00:35:20.680 --> 00:35:24.940
I've written this expression
"matrix of second derivatives."
00:35:24.940 --> 00:35:27.165
What's the matrix of
second derivatives?
00:35:29.690 --> 00:35:31.970
That's one second
derivative now,
00:35:31.970 --> 00:35:40.020
but if I'm in two dimensions,
I have a two by two matrix,
00:35:40.020 --> 00:35:46.480
it's the second x derivative,
the second x derivative goes
00:35:46.480 --> 00:35:49.070
there --
00:35:49.070 --> 00:35:56.360
shall I write it --
fxx, if you like, fxx,
00:35:56.360 --> 00:36:00.170
that means the second derivative
of f in the x direction.
00:36:00.170 --> 00:36:04.500
fyy, second derivative
in the y direction.
00:36:04.500 --> 00:36:07.970
Those are the pure derivatives,
second derivatives.
00:36:07.970 --> 00:36:10.950
They have to be positive.
00:36:10.950 --> 00:36:13.430
For a minimum.
00:36:13.430 --> 00:36:15.590
This number has to be
positive for a minimum.
00:36:15.590 --> 00:36:17.730
That number has to be
positive for a minimum.
00:36:17.730 --> 00:36:20.410
But, that's not enough.
00:36:20.410 --> 00:36:23.680
Those numbers have to
somehow be big enough
00:36:23.680 --> 00:36:30.090
to overcome this
cross-derivative,
00:36:30.090 --> 00:36:31.780
Why is the matrix symmetric?
00:36:31.780 --> 00:36:35.730
Because the second derivative
f with respect to x and y is
00:36:35.730 --> 00:36:37.950
equal to --
00:36:37.950 --> 00:36:42.170
I can, that's the beautiful fact
about second derivatives, is
00:36:42.170 --> 00:36:46.480
I can do those in either order
and I get the same thing.
00:36:46.480 --> 00:36:54.410
So this is the same as
that, and so, that's
00:36:54.410 --> 00:36:57.300
the matrix of
second derivatives.
00:36:57.300 --> 00:37:01.280
And the test is, it has
to be positive definite.
00:37:01.280 --> 00:37:05.960
You might remember,
from, tucked in somewhere
00:37:05.960 --> 00:37:08.990
near the end of eighteen o'
two or at least in the book,
00:37:08.990 --> 00:37:13.590
was the condition for a
minimum, For a function
00:37:13.590 --> 00:37:15.470
of two variables.
00:37:15.470 --> 00:37:17.180
Let's -- when do
you have a minimum?
00:37:17.180 --> 00:37:19.740
For a function of two
variables, believe me,
00:37:19.740 --> 00:37:24.200
that's what Calculus is for.
00:37:24.200 --> 00:37:29.270
The condition is first
derivatives have to be zero.
00:37:29.270 --> 00:37:32.630
And the matrix of
second derivatives
00:37:32.630 --> 00:37:35.280
has to be positive definite.
00:37:35.280 --> 00:37:39.650
So you maybe remember there
was an fxx times an fyy that
00:37:39.650 --> 00:37:42.610
had to be bigger than
an an fxy squared,
00:37:42.610 --> 00:37:46.630
that's just our
determinant, two by two.
00:37:46.630 --> 00:37:51.480
But now, we now know the
answer for three by three,
00:37:51.480 --> 00:37:57.770
m by m, because we can do
elimination by m by m matrices,
00:37:57.770 --> 00:38:01.660
we can connect eigenvalues
of m by m matrices,
00:38:01.660 --> 00:38:06.070
we can do sum of squares, sum
of m squares instead of only two
00:38:06.070 --> 00:38:11.470
squares; and so
let's take a, let
00:38:11.470 --> 00:38:14.000
me go over here to do a
three by three example.
00:38:17.110 --> 00:38:18.510
So, three by three example.
00:38:23.445 --> 00:38:23.945
OK.
00:38:27.060 --> 00:38:29.113
Oh, let me --
00:38:32.980 --> 00:38:34.670
shall I use my favorite matrix?
00:38:37.210 --> 00:38:39.080
You've seen this matrix before.
00:38:45.340 --> 00:38:49.680
Yes, let's use the good matrix,
four by one, oops, open.
00:38:56.130 --> 00:38:57.950
Is that matrix
positive definite?
00:39:01.300 --> 00:39:04.440
What's -- so I'm going to ask
questions about this matrix,
00:39:04.440 --> 00:39:08.260
is it positive
definite, first of all?
00:39:08.260 --> 00:39:10.970
What's the function
associated with that matrix,
00:39:10.970 --> 00:39:12.780
what's the x transpose Ax?
00:39:16.260 --> 00:39:18.860
Is -- do we have a
minimum for that function,
00:39:18.860 --> 00:39:19.865
at zero?
00:39:22.460 --> 00:39:25.380
And then even
what's the geometry?
00:39:25.380 --> 00:39:26.060
OK.
00:39:26.060 --> 00:39:29.200
First of all, is the
matrix positive definite,
00:39:29.200 --> 00:39:33.460
now I've given you the
numbers there so you can take
00:39:33.460 --> 00:39:35.370
the determinants, maybe
that's the quickest,
00:39:35.370 --> 00:39:36.910
is that what you
would do mentally,
00:39:36.910 --> 00:39:39.660
if I give you all a
matrix on a quiz and say
00:39:39.660 --> 00:39:43.010
is it positive definite or not?
00:39:43.010 --> 00:39:47.240
I would take that determinant
and I'd give the answer two.
00:39:47.240 --> 00:39:49.130
I would take that
determinant and I
00:39:49.130 --> 00:39:55.380
would give the answer for
that two by two determinant,
00:39:55.380 --> 00:39:57.400
I'd give the answer
three, and anybody
00:39:57.400 --> 00:40:04.220
remember the answer for the
three by three determinant?
00:40:04.220 --> 00:40:08.220
It was four, you remember
for these special matrices,
00:40:08.220 --> 00:40:13.190
when we do determinants, they
went up two, three, four, five,
00:40:13.190 --> 00:40:15.950
six, they just went up linearly.
00:40:15.950 --> 00:40:22.940
So that matrix has -- the
determinants are two, three,
00:40:22.940 --> 00:40:24.615
and four.
00:40:24.615 --> 00:40:25.115
Pivots.
00:40:27.630 --> 00:40:31.160
What are the pivots
for that matrix?
00:40:31.160 --> 00:40:34.470
I'll tell you, they're --
the first pivot is two,
00:40:34.470 --> 00:40:37.520
the next pivot is
three over two,
00:40:37.520 --> 00:40:39.730
the next pivot is
four over three.
00:40:42.590 --> 00:40:45.170
Because, the product
of the pivots
00:40:45.170 --> 00:40:47.550
has to give me
those determinants.
00:40:47.550 --> 00:40:50.370
The product of these two pivots
gives me that determinant;
00:40:50.370 --> 00:40:53.574
the product of all the pivots
gives me that determinant.
00:40:53.574 --> 00:40:54.615
What are the eigenvalues?
00:41:03.020 --> 00:41:05.717
Oh, I don't know.
00:41:08.600 --> 00:41:12.860
The eigenvalues I've got, what
do I have a cubic equation --
00:41:12.860 --> 00:41:14.320
a degree three equation?
00:41:17.230 --> 00:41:19.350
There are three
eigenvalues to find.
00:41:23.350 --> 00:41:25.200
If I believe what
I've said today,
00:41:25.200 --> 00:41:27.800
what do I know about
these eigenvalues,
00:41:27.800 --> 00:41:30.710
even though I don't
know the exact numbers.
00:41:30.710 --> 00:41:34.535
I -- I remember the numbers.
00:41:37.100 --> 00:41:39.740
Because these matrices
are so important
00:41:39.740 --> 00:41:43.750
that people figure them.
00:41:43.750 --> 00:41:47.440
But -- what do you believe
to be true about these three
00:41:47.440 --> 00:41:52.950
eigenvalues -- you believe
that they are all positive.
00:41:52.950 --> 00:41:54.270
They're all positive.
00:41:54.270 --> 00:42:00.910
I think that they are two
minus square root of two, two,
00:42:00.910 --> 00:42:02.630
and two plus the
square root of two.
00:42:02.630 --> 00:42:03.720
I think.
00:42:03.720 --> 00:42:05.490
Let me just --
00:42:05.490 --> 00:42:08.280
I can't write those numbers
down without checking
00:42:08.280 --> 00:42:10.870
the simple checks, what
the first simple check is
00:42:10.870 --> 00:42:15.110
the trace, so if I add
those numbers I get six
00:42:15.110 --> 00:42:18.970
and if I add those
numbers I get six.
00:42:18.970 --> 00:42:24.150
The other simple test is
the determinant, if I --
00:42:24.150 --> 00:42:26.930
can you do this, can you
multiply those numbers
00:42:26.930 --> 00:42:29.240
together?
00:42:29.240 --> 00:42:32.610
I guess we can multiply
by two out there.
00:42:32.610 --> 00:42:35.010
What's two minus square
root of two times two
00:42:35.010 --> 00:42:39.530
plus square root of two,
that'll be four minus two,
00:42:39.530 --> 00:42:41.610
that'll be two,
yeah, two times two,
00:42:41.610 --> 00:42:46.280
that's got the determinant,
right, so it's got,
00:42:46.280 --> 00:42:49.810
it's got a chance of being
correct and I think it is.
00:42:49.810 --> 00:42:51.990
Now, what's the x transpose Ax?
00:42:51.990 --> 00:42:54.610
I better give myself
enough room for that.
00:42:54.610 --> 00:42:59.180
x transpose Ax for this guy.
00:42:59.180 --> 00:43:07.290
It's two x1 squareds, and two x2
squareds, and two x3 squareds.
00:43:07.290 --> 00:43:10.800
Those come from the
diagonal, those were easy.
00:43:10.800 --> 00:43:13.490
Now off the diagonal
there's a minus and a minus,
00:43:13.490 --> 00:43:20.010
they come together there'll be
a minus two minus two whats?
00:43:20.010 --> 00:43:26.390
Are coming from this one two and
two one position, is the x1 x2.
00:43:26.390 --> 00:43:30.900
I'm doing mentally
a multiplication
00:43:30.900 --> 00:43:34.300
of this matrix
times a row vector
00:43:34.300 --> 00:43:37.680
on the left times a column
vector on the right,
00:43:37.680 --> 00:43:41.730
and I know that these numbers
show up in the answer.
00:43:41.730 --> 00:43:45.060
The diagonal is
the perfect square,
00:43:45.060 --> 00:43:49.110
this off diagonal is
a minus two x1 x2,
00:43:49.110 --> 00:43:55.280
and there are no x1 x3-s, and
there're minus two x2 x3-s.
00:43:55.280 --> 00:43:58.860
And I believe that that
expression is always positive.
00:44:01.670 --> 00:44:06.010
I believe that that
curve, that graph, really,
00:44:06.010 --> 00:44:09.720
of that function,
this is my function f,
00:44:09.720 --> 00:44:14.770
and I'm in more dimensions
now than I can draw, it --
00:44:14.770 --> 00:44:20.080
but the graph of that
function goes upwards.
00:44:20.080 --> 00:44:22.320
It's a bowl.
00:44:22.320 --> 00:44:26.320
Or maybe the right word is --
00:44:26.320 --> 00:44:34.210
just forgot, what's
a long word for bowl?
00:44:34.210 --> 00:44:42.570
Hm, maybe paraboloid, I
think, paraboloid comes in.
00:44:42.570 --> 00:44:46.150
I'll edit the tape
and get that word in.
00:44:46.150 --> 00:44:54.410
Bowl, let's say, is, that,
so that, and if I can --
00:44:54.410 --> 00:44:56.700
I could complete the
squares, I could write that
00:44:56.700 --> 00:44:59.740
as the sum of three squares,
and those three squares
00:44:59.740 --> 00:45:02.780
would get multiplied
by the three pivots.
00:45:02.780 --> 00:45:05.440
And the pivots are all positive.
00:45:05.440 --> 00:45:08.760
So I would have positive
pivots times squares,
00:45:08.760 --> 00:45:11.630
the net result would
be a positive function
00:45:11.630 --> 00:45:14.520
and a bowl which goes upwards.
00:45:14.520 --> 00:45:19.230
And then, finally, if I cut --
if I slice through this bowl,
00:45:19.230 --> 00:45:23.730
if I -- now I'm asking you
to stretch your visualization
00:45:23.730 --> 00:45:26.680
here, because I'm
in four dimensions,
00:45:26.680 --> 00:45:33.610
I've got x1 x2 x3 in the base,
and this function is z, or f,
00:45:33.610 --> 00:45:35.000
or something.
00:45:35.000 --> 00:45:38.320
And its graph is going up.
00:45:38.320 --> 00:45:40.530
But I'm in four dimensions,
because I've got three
00:45:40.530 --> 00:45:43.410
in the base and then
the upward direction,
00:45:43.410 --> 00:45:49.640
but now if I cut through this
four-dimensional picture,
00:45:49.640 --> 00:45:55.720
at level one, so, suppose
I cut through this thing
00:45:55.720 --> 00:45:58.100
at height one.
00:45:58.100 --> 00:46:01.470
So I take all the points
that are at height one.
00:46:04.980 --> 00:46:07.770
That gives me --
00:46:07.770 --> 00:46:14.180
it gave me an ellipse over
there, in that two by two case,
00:46:14.180 --> 00:46:19.350
in this case, this will be
the equation of an ellipsoid,
00:46:19.350 --> 00:46:20.830
a football in other words.
00:46:23.770 --> 00:46:25.170
Well, not quite a football.
00:46:25.170 --> 00:46:26.450
A lopsided football.
00:46:26.450 --> 00:46:30.960
What would be, can I try
to describe to you what
00:46:30.960 --> 00:46:35.590
the ellipsoid will look
like, this ellipsoid,
00:46:35.590 --> 00:46:40.440
I'm sorry that the, that I've
ended the matrix right --
00:46:40.440 --> 00:46:46.230
at the point, let's -- let me be
sure you've seen the equation.
00:46:46.230 --> 00:46:51.130
Two x1 squared, two x2 squared,
two x3 squared, minus two
00:46:51.130 --> 00:46:57.000
of the cross parts, equal what?
00:46:57.000 --> 00:47:00.720
That is the equation
of a football, so what
00:47:00.720 --> 00:47:03.780
do I mean by a football
or an ellipsoid?
00:47:03.780 --> 00:47:12.520
I mean that, well,
I'll draw a few.
00:47:12.520 --> 00:47:21.750
It's like that,
it's got a center,
00:47:21.750 --> 00:47:31.990
and it's got it's got
three principal directions.
00:47:31.990 --> 00:47:32.830
This ellipsoid.
00:47:32.830 --> 00:47:35.660
So -- you see what I'm saying,
if we have a sphere then all
00:47:35.660 --> 00:47:36.890
directions would be the same.
00:47:39.530 --> 00:47:44.750
If we had a true football, or
it's closer to a rugby ball,
00:47:44.750 --> 00:47:48.110
really, because it's more
curved than a football,
00:47:48.110 --> 00:47:52.270
it would have one long
direction and the other two
00:47:52.270 --> 00:47:54.020
would be equal.
00:47:54.020 --> 00:47:56.240
That would be like
having a matrix
00:47:56.240 --> 00:47:59.700
that had one
eigenvalue repeated.
00:47:59.700 --> 00:48:02.080
And then one other different.
00:48:02.080 --> 00:48:04.940
So this sphere comes from,
like, the identity matrix,
00:48:04.940 --> 00:48:07.600
all eigenvalues the same.
00:48:07.600 --> 00:48:12.370
Our rugby ball comes
from a case where --
00:48:12.370 --> 00:48:17.070
three, the three, two of the
three eigenvalues are the same.
00:48:17.070 --> 00:48:20.000
But we know how the case
where -- the typical case,
00:48:20.000 --> 00:48:23.340
where the three eigenvalues
were all different.
00:48:23.340 --> 00:48:25.551
So this will have --
00:48:28.440 --> 00:48:31.830
How do I say it, if I look
at this ellipsoid correctly,
00:48:31.830 --> 00:48:37.200
it'll have a major axis,
it'll have a middle axis,
00:48:37.200 --> 00:48:41.520
and it'll have a minor axis.
00:48:41.520 --> 00:48:44.940
And those three axes
will be in the direction
00:48:44.940 --> 00:48:47.450
of the eigenvectors.
00:48:47.450 --> 00:48:50.760
And the lengths of
those axes will be
00:48:50.760 --> 00:48:52.496
determined by the eigenvalues.
00:48:55.260 --> 00:48:56.770
I can get --
00:48:56.770 --> 00:49:02.520
turn this all into linear
algebra, because we have --
00:49:02.520 --> 00:49:06.410
the right thing we know about
eigenvectors and eigenvalues,
00:49:06.410 --> 00:49:08.120
for that matrix is what?
00:49:08.120 --> 00:49:11.970
Of -- let me just tell you that,
repeat the main linear algebra
00:49:11.970 --> 00:49:13.070
point.
00:49:13.070 --> 00:49:18.970
How could we turn what
I said into algebra;
00:49:18.970 --> 00:49:25.050
we would write this A as
Q, the eigenvector matrix,
00:49:25.050 --> 00:49:30.370
times lambda, the eigenvalue
matrix times Q transposed.
00:49:30.370 --> 00:49:33.010
The principal axis
theorem, we'll call it,
00:49:33.010 --> 00:49:33.900
now.
00:49:33.900 --> 00:49:37.980
The eigenvectors tell us the
directions of the principal
00:49:37.980 --> 00:49:39.110
axes.
00:49:39.110 --> 00:49:43.240
The eigenvalues tell us the
lengths of those axes, actually
00:49:43.240 --> 00:49:45.160
the lengths, or
the half-lengths,
00:49:45.160 --> 00:49:48.970
or one over the
eigenvalues, it turns out.
00:49:48.970 --> 00:49:53.260
And that is the
matrix factorization
00:49:53.260 --> 00:49:55.490
which is the most
important matrix
00:49:55.490 --> 00:50:00.540
factorization in our
eigenvalue material so far.
00:50:00.540 --> 00:50:04.320
That's diagonalization
for a symmetric matrix,
00:50:04.320 --> 00:50:09.350
so instead of the inverse
I can write the transposed.
00:50:09.350 --> 00:50:09.930
OK.
00:50:09.930 --> 00:50:14.140
I've -- so what I've tried
today is to tell you the --
00:50:14.140 --> 00:50:20.510
what's going on with
positive definite matrices.
00:50:20.510 --> 00:50:23.020
Ah, you see all how all
these pieces are there
00:50:23.020 --> 00:50:25.721
and linear algebra
connects them.
00:50:25.721 --> 00:50:26.220
OK.
00:50:28.760 --> 00:50:30.680
See you on Friday.